Revisited analysis of gas convection and heat transfer in micro channels: Influence of viscous stress power at wall on Nusselt number

This paper deals with the modeling of weakly rarefied and dilute gas flows in micro channels by the continuum approach, valid for Knudsen numbers smaller than about 0.1. It particularly focuses on the modeling of the associated heat transfer. The models proposed in the literature for the forced convection of gas flows in long micro channels between two infinite plates are more specifically discussed. The complete model for such flows is reminded after a compilation and a brief description of their possible applications in industries. The compatibility of the pressure work and viscous dissipation in the energy equation with the power of the viscous stress at the walls is discussed in detail. A dimensional analysis is proposed in the context of long micro channels. Analytical solutions for the velocity and temperature fields and for the Nusselt number are provided in the case of compressible micro-flows in isothermally heated flat plate channels, with pressure work and viscous dissipation included. The choice of an appropriate Nusselt number, including the power of the viscous stress at the wall, is particularly discussed. It is shown analytically and numerically, by solving the complete model for an isothermal wall micro-channel, that the Nusselt number tends to zero when the hydraulic diameter decreases, that is when the Reynolds number decreases and the Knudsen number increases. This could theoretically explain the very small values of the Nusselt number obtained in the experiments by Demsis et al. (2009, 2010).